---
source_pdf: Sector_Resolved_Measurement_in_a_Scalar_Refractive_Gravity_Framework.pdf
title: "Sector-Resolved Measurement in a Scalar Refractive Gravity Framework:"
site: https://densityfielddynamics.com/
author: Gary Alcock
framework: Density Field Dynamics (DFD)
format_note: "Markdown extracted from the source PDF for clean AI ingestion. No editorial changes; mathematical typography in the PDF is authoritative."
---

Sector-Resolved Measurement in a Scalar Refractive Gravity Framework:
Unique Geometry, Pointer Fixing, and a Laboratory Discriminator
Gary Alcock
Los Angeles, CA, USA
(Dated: October 2, 2025)
We show that in a scalar refractive gravity framework (a “ψ-field” replacing curved spacetime),
superposed mass distributions source one geometry, resolving the Penrose paradox of multiple spacetimes. Quantum evolution remains unitary under a self-adjoint Hamiltonian in that geometry. Because cavity photons and atomic transitions couple unequally to ψ (cavities track c1 = c e−ψ ; atomic
transitions either cancel leading ψ contributions or acquire a uniform fractional shift), meter states
acquire ψ-dependent phases while the system does not (or does so differently), fixing the pointer
basis and driving decoherence by environment coupling. We emphasize: we do not derive outcome
selection or the Born rule. The concrete corollary is a laboratory test:
∆Φ
∆R
=ξ 2 ,
R
c

ξ ≡ αph − αat ,

where αph and αat parameterize photon and atomic ψ-response at leading order. General Relativity
(GR) implies ξ = 0 (sector equality); any ψ-framework with sector asymmetry implies ξ ̸= 0. On
Earth this lever arm is 1.1 × 10−14 per 100 m — within reach of modern metrology.

NOTATION

We use δψ for infinitesimal/local changes (e.g. perturbation theory at a single location) and ∆ψ for finite differences between two altitudes. The gravitational potential difference satisfies ∆ψ = 2 ∆Φ/c2 .

INTRODUCTION

The quantum measurement problem has resisted
a fully satisfactory resolution. Standard approaches
(Copenhagen, many-worlds, de Broglie–Bohm) either invoke collapse, branches, or nonlocal hidden variables.
Penrose sharpened this by pointing out that a quantum superposition of mass distributions should gravitationally source multiple geometries, which is inconsistent
with a single wavefunction evolving unitarily [1, 2].
In a scalar refractive gravity approach, one posits a
universal refractive index n = eψ(x) such that the oneway speed of light is c1 (x) = c e−ψ(x) . Matter acceler2
ations follow a = c2 ∇ψ. The field obeys a nonlinear
sourcing equation,




8πG
∇ · µ |∇ψ|/a⋆ ∇ψ = − 2 ρ − ρ̄ ,
c
h

i

(1)

with a monotone crossover function µ. Under standard
conditions for quasilinear elliptic PDEs, this has a unique
solution for a given source. The key claim becomes: any
superposition sources exactly one ψ.
We develop the following chain:
(1) Superpositions → one ψ geometry.
(2) Quantum evolution in that ψ background is unitary.
(3) Cavities (photons) and atoms can respond differently
to ψ.

ρL

ψ = |a|2 ψL + |b|2 ψR

ρR

FIG. 1. Superposed densities source a single refractive field
ψ; this mirrors how quantum expectation values source classical fields, e.g. ⟨j µ ⟩ sourcing electromagnetism in semiclassical
QED.

(4) This sectoral asymmetry pins the pointer basis; under
generic environment coupling, the system decoheres in
that basis.
(5) The falsifiable corollary: ∆R/R = ξ ∆Φ/c2 with ξ =
αph − αat .
(6) We do not derive the Born rule or single-outcome
selection — that remains interpretational.

ONE ψ FOR SUPERPOSITIONS

Let ρ̂(x) be the mass density operator in the quantum
state |Ψ⟩. Define the effective classical source
ρeff (x) = ⟨Ψ|ρ̂(x)|Ψ⟩.
If |Ψ⟩ = a|L⟩+b|R⟩, then ρeff ≈ |a|2 ρL +|b|2 ρR . Inserting
this into Eq. (1) yields a unique ψ (given monotonic µ),
with no geometric branching.
Thus the Penrose objection (superpositions ⇒ multiple geometries) is neutralized: only one ψ ever exists,
determined by the expectation density.

2
Cavity photons and ψ

QUANTUM EVOLUTION IN A ψ BACKGROUND

For a rigid cavity of length L and longitudinal mode
index q ∈ N, the resonance is

The single-particle Hamiltonian is
Ĥ = −



ℏ2
∇· e−ψ ∇ + mΦ,
2m

Φ=−

c2
ψ,
2

(2)

which is self-adjoint with respect to the natural inner
product; evolution iℏ∂t |Ψ⟩ = Ĥ|Ψ⟩ is unitary with a conserved probability current.

Atomic ψ-response: constant vs gradient effects

Tiny gradient effects. On Earth, ∇ψ = 2g/c2 ≈ 2.2 ×
10−16 m−1 . Over an atomic size a0 ∼ 5 × 10−11 m, the
variation δψ ≲ 10−26 . Perturbations from spatial variation (both in the kinetic operator and in mΦ(x)) therefore produce level-dependent fractional shifts ≲ 10−26 ,
utterly negligible compared to the ∼ 10−14 altitude lever
arm of our experiment.
Constant ψ0 : uniform fractional scaling (hydrogenic
example). Write ψ(x) = ψ0 + δψ(x) with δψ neglected.
Then


c2
ℏ2 2
−ψ0
Ĥ ≈ e
∇ + VEM + mΦ0 , Φ0 = − ψ0 .
−
2m
2
The common shift mΦ0 cancels in all transition frequencies. Treat the kinetic prefactor as a small perturbation
ℏ2
δ Ĥ = −ψ0 T̂ with T̂ = − 2m
∇2 . For Coulomb binding,
the virial theorem gives ⟨T ⟩n = −En (with En < 0).
First-order perturbation theory yields
δEn = ⟨n|δ Ĥ|n⟩ = −ψ0 ⟨T ⟩n = ψ0 En ,
so each bound energy acquires the same fractional shift
δEn /En = ψ0 . For transitions,

fcav =

q c1
,
2L

c1 = c e−ψ .

Thus
δfcav
= − δψ
fcav

⇒

αph = −1,

assuming L is fixed by rigid body mechanics at the relevant precision (elastic/thermal effects bounded as systematics).

Sector coefficients and what the experiment
measures

Summarizing the leading-order fractional responses for
a small local change δψ:
δfcav
= αph δψ,
fcav

δfat
= αat δψ.
fat

In the kinetic-only hydrogenic estimate above, αat = +1.
If electromagnetic material parameters co-vary with ψ
within solids and atoms such that internal energies receive compensating factors, αat can be suppressed (αat ≃
0 in an ideal nondispersive cancellation). Our proposed
observable is the difference
ξ ≡ αph − αat .
GR enforces sector equality and hence ξ = 0. Any inequivalence gives ξ ̸= 0. The experiment measures ξ
directly.

δωab
δ(Ea − Eb )
ψ0 (Ea − Eb )
≡
=
= ψ0 ,
ωab
Ea − Eb
Ea − E b

POINTER BASIS: OPERATIONAL
(METER-CHOSEN)

confirming uniformity for transition frequencies at leading order. We denote this atomic leading-order coefficient
by αat = +1 in this minimal, kinetic-only model.1

Let system S (atom) and meter M (cavity photons)
interact and then couple to the environment E. The
total Hamiltonian reads
Ĥtot = ĤS [ψ] + ĤM [ψ] + ĤSM + ĤM E .

1 More general atoms (non-Coulombic potentials, relativistic and

QED corrections) preserve the conclusion that a constant multiplicative change of the kinetic operator induces a uniform firstorder fractional shift across transitions, modulo small statedependent corrections; the uniformity follows from virial-type
relations for homogeneous potentials. If electromagnetic parameters co-vary with ψ in matter such that VEM picks up compensating factors, the net αat can be reduced or nulled (αat ≃ 0);
the experiment measures ξ = αph − αat .

Because ĤM imprints ψ-dependent phases on meter
states with coefficient αph while ĤS imprints with (possibly different) αat , the meter determines the pointer basis
in which decoherence occurs. If one instead engineered
an atomic (matter-sector) meter, the pointer basis would
follow the corresponding atomic eigenstates. This is fully
standard in environment-induced superselection: the apparatus/environment coupling selects the basis.

3
INFLUENCE FUNCTIONAL PROOF OF
DECOHERENCE

∆R/R

We give the reduced density matrix evolution explicitly.
Setup. Initial state
X
|Ψ0 ⟩ =
ci |si ⟩ ⊗ |m0 ⟩ ⊗ |E0 ⟩.
100

i

Under unitary evolution U (t) generated by Ĥtot , the state
at time t can be written (in a standard pre-measurement
model) as
X
|Ψ(t)⟩ =
ci |si ⟩ ⊗ |mi (t)⟩ ⊗ |Ei (t)⟩,

200

ξ ̸= 0 (sector asymmetry)
GR (ξ = 0)
(m)
300 ∆h400

FIG. 2. Predicted cavity–atom slope: null (GR, ξ = 0) vs
non-null (ξ ̸= 0). Linear-gradient model is valid over ≲ 400 m;
curvature corrections are < 10−18 and negligible at present
precision.

i

Between two altitudes separated by ∆h on Earth,
(i)

where |mi (t)⟩ = e−iĤM t/ℏ |m0 ⟩ and |Ei (t)⟩ incorporate
the M –E interactions conditioned on branch i.
Reduced state and influence functional. Tracing out
M E,
∗
ρij
S (t) = ci cj Fij (t),

Fij (t) ≡ ⟨mj (t)|mi (t)⟩ ⟨Ej (t)|Ei (t)⟩.

Because ĤM depends on ψ with coefficient αph while ĤS
depends with αat , distinct branches i ̸= j accumulate
different meter phases
o
ni Z t
(ij)
dt′ ∆HM [ψ(t′ )] ,
⟨mj (t)|mi (t)⟩ ∼ exp
ℏ 0
(ij)

with ∆HM proportional to (αph −αat ) δψ through the
S–M coupling. Coupling to E induces damping of offdiagonal terms; in the Born–Markov limit one obtains


|⟨Ej (t)|Ei (t)⟩| ≈ exp −Γt ,
for some decoherence rate Γ set by M –E couplings.
Hence
|Fij (t)| → 0

(i ̸= j),

and
ρS (t) ≈

X

|ci |2 |si ⟩⟨si |.

i

We stress: this does not derive selection of one outcome
or the Born rule; it shows suppression of interference in
a basis fixed by sector response and apparatus coupling.
EXPERIMENTAL COROLLARY: ALTITUDE
SLOPE

At a single location, a small local change δψ gives
δR
δ(fcav /fat )
≡
= (αph − αat ) δψ ≡ ξ δψ.
R
(fcav /fat )

∆ψ =

2 ∆Φ
2 g∆h
≈
,
c2
c2

so the finite change is
∆Φ
∆R
=ξ 2 .
R
c
With g∆h/c2 ≈ 1.1 × 10−14 per 100 m, a nonzero ξ produces a clean, linear slope; ξ = 0 gives a strict null. Modern cavities (stability 10−16 –10−17 ) and optical clocks
(< 10−18 ) make this test feasible.

RELATION TO EXISTING EXPERIMENTS AND
WHY THIS IS UNTESTED

High-precision redshift tests comparing atom vs atom
clocks have reached fractional 10−17 over 33–40 cm [3].
Transportable optical lattice clocks have compared sites
separated by ∼ 100–450 m [4]. Ultra-stable cavities
(room-temperature and cryogenic) reach 10−16 –10−17
linewidths and underpin state-of-the-art optical clocks
[5, 6]. Searches for ultralight dark matter also compare
cavity and atomic references, but probe temporal modulations, not an altitude slope [7].
To our knowledge, a stationary, two-altitude, sectorresolved measurement of fcav /fat reporting a geometrylocked slope at the 10−14 /100 m level has not been published. This likely reflects practical priorities (atomicto-atomic comparisons for timekeeping) rather than impossibility: the required components are standard. A
dedicated protocol with dispersion bounds, orientation
flips, and hardware swaps would decisively determine ξ.

CONCLUSION

This framework eliminates the Penrose paradox by enforcing one geometry, shows how the pointer basis is fixed

4
by sector response (operationally, by the meter), and reduces a conceptual tension to a falsifiable, binary discriminator. We make no claim to derive outcome selection. The experiment measures ξ = αph − αat via
a height-dependent slope of fcav /fat at ∼ 10−14 /100 m.
Independent of whether αat ≈ 0 (material co-variance) or
αat ≈ +1 (kinetic-only scaling), the key discriminator is
ξ: any ξ ̸= 0 signals physics beyond GR’s sector equality.

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doi:10.1007/BF02105068

[2] R. Penrose, “On the Gravitization of Quantum Mechanics
1: Quantum State Reduction,” Found. Phys. 44, 557–575
(2014). doi:10.1007/s10701-013-9770-0
[3] C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland,
“Optical Clocks and Relativity,” Science 329, 1630–1633
(2010). doi:10.1126/science.1192720
[4] M. Takamoto et al., “Test of General Relativity by a Pair
of Transportable Optical Lattice Clocks,” Nat. Photonics
14, 411–415 (2020). doi:10.1038/s41566-020-0619-8
[5] T. Kessler et al., “A sub-40-mHz-linewidth laser based on
a silicon single-crystal optical cavity,” Nat. Photonics 6,
687–692 (2012). doi:10.1038/nphoton.2012.217
[6] D. G. Matei et al., “1.5 µm Lasers with Sub-10
mHz Linewidth,” Phys. Rev. Lett. 118, 263202 (2017).
doi:10.1103/PhysRevLett.118.263202
[7] C. J. Kennedy et al., “Precision Metrology Meets Cosmology: Measuring Dark Matter with Atomic Clocks,” Nat.
Phys. 16, 112–117 (2020). doi:10.1038/s41567-019-0719-8